IS RATE ADAPTATION BENEFICIAL FOR INTER-SESSION NETWORK CODING?

Yuchul Kim and Gustavo de VecianaWireless Networking and Communications Group

Departement of Electrical and Computer EngineeringThe University of Texas at Austin

Email: {yuchu1.kim@mail,gustavo@ece}.utexas.edu

Abstract-This paper considers the interplay between rate adap­tation and inter-session network coding gains in wireless ad hocor mesh networks. Inter-session network coding opportunities atrelay nodes depend on packets being overheard by surroundingnodes - the more packets nodes overhear, the more opportunitiesrelays have to combine packets, resulting in a potential increasein network throughput. Thus, by adapting its transmission rate, anode can increase the range over which its packets are overheard,enabling additional opportunities for coding and increased overallthroughput. This paper considers inter-session coding, restricted to asingle relay (bottleneck) node, or star topology. Even for such simpletopologies the optimal joint rate adaptation and network codingpolicy is known to be NP-hard, so we consider an optimal pairwisecoding policy which can be formulated as a linear programming andprovide a simple heuristic for rate adaptation for network coding. Weevaluate the averaged throughput in two different scenarios, in whichrelays have different access opportunities, giving some intuition onthe impact of rate adaption in lightly and heavily loaded systems.The gains of joint rate adaptation and network coding are marginalwhen relay has higher access opportunity than other nodes, or whenMAC operates ideally. The gain ranges from 9°,/c) to 190/0 compared toa network without network coding and is around 4% over a networkusing regular network coding. While, when the relay has equal accessopportunity as other source nodes, which is more typical of todaysMAC protocols under heavy loads, the gain ranges from 40°,/c) to62% as compared to standard relaying case and is upto around 20%as compared to a network with regular pairwise network coding.And we further increase this gain from 40% to 120°,/c) by replacingpairwise coding with sub-optimal general network coding scheme.

I. INTRODUCTION

Since Ahlswede's seminal work [1], network coding has re­ceived significant attention. It has since been shown that networkcoding achieves maximum capacity for multicast sessions inwired networks while increasing the reliability of lossy networks[2]-[6]. Most work to date has focused on, intra-session networkcoding, where only packets in the same session are encodedtogether, e.g., [4]-[6]. However, the work of Katti and Katabiwhich proposed the scheme called COPE, does allow codingacross different sessions or inter-session network coding [7]. Theyobserve that overheard packets, in the context of broadcast wire­less media, can be effectively exploited to enable network codingresulting in further throughput improvements. This was followedby [8]-[10] where efforts were made to quantify the possible gainsin general wireless networks; they show the expected gains are atmost a constant factor, e.g., up to around 2 or 3. However, thesebounds were found in an analytical framework which is quite

This work of Yuchul Kim is supported by a grant of the CISCO researchfoundation.

978-1-4244-2677-5/08/$25.00 @2008 IEEE

idealized, and it is not yet known how much we can actuallyexpect from network coding in practical settings.

In this paper, we explore the potential gains of joint rate adap­tation and network coding on random star network topologies.The star topology network with single relay and multiple sourceand destination nodes can be viewed as a basic building block forgeneral wireless networks. The key idea is to transmit at a reducedrate so as to allow additional neighboring nodes to overhear trans­missions creating additional opportunities for network coding ata relay node. To this end, we propose a polynomial time heuristicto jointly determine sub-optimal Tx rates and complementaryinter-session coding assuming only pairwise coding. We studythe performance of the proposed scheme in two regimes withdifferent assumptions, on MAC contention and in particular onthe relay node's access to the medium. It turns out that joint rateadaptation and network coding is effective when the relay has aequal access opportunity to the medium as other nodes, or whenthe relay is congested due to its limited access opportunity.

The rest of this paper is organized as follows. In Section II,we develop the key intuition for coding aware rate adaptationand its potential gain in terms of increased coverage area andrate region. We then formally describe a system and formulatethe optimal pariwise coding problem in Section III. In SectionI~ we propose a polynomial time heuristic algorithm for sub­optimal rate adaptation. The proposed algorithm is evaluated inSection V. And, we conclude in Section VI.

II. CODING AWARE RATE ADAPTATION

A. Inter-session network coding

We begin by briefly summarizing the COPE approach [7]. In anetwork with COPE, destination nodes overhear packet transmis­sions from neighboring nodes and store them. Information aboutoverheard packets is subsequently sent to neighboring relay nodes.Using this information a relay node combines packets such thatthe intended destination nodes can decode them. When destinationnodes receive a combined packet, they can extract the desiredpackets from the combined packet using the packets they havepreviously overheard. This approach is effective at increasingthroughput and alleviating congestion in relay nodes acting astraffic 'hubs'.

B. Intuition underlying inter-session network coding in wirelessenvironments

Note that inter-session coding opportunities at relay nodesdepend on the packets overheard by surrounding nodes. That

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CRnG!,..__-( R;/' ,

'CBCeRC ,,

c c c

Fig. I: Three simple toy networks where A sends a packet to C and B sends a packet to A through R:(a) C's overhearing B comes for free, (b) capacity can be increased with ratereduction from C B R to C Be, and (c) rate reduction degrades network perfonnance.

is, if nodes overhear more packets, a relay may have more op­portunities for combining packets, potentially increasing networkperformance. Until now, researchers have assumed that only nodesin the transmission (Tx) range of the transmitter could overhearpackets. So, node placement and the Tx range determined thepossibility of such overhearing. However, we argue that one canincrease coding opportunities by dynamically increasing the Txrange so that more nodes can overhear a transmission. This canbe achieved by increasing the Tx power or reducing the Tx rate.Increasing the Tx power l may result in a reduction of spatialreuse, so it is not clear if one can realize increased performance.However by decreasing the Tx rate, one can increase the Tx rangewhile keeping interference power at the same level as before. Sothe key insight in this paper can be summarized as follows:

A node reducing its instantaneous Tx rate can increase itsoverhearing range, leading to possible increase in network codingopportunities and thus in network throughput.

This statement appears counter intuitive since reducing Tx ratewould typically decrease network throughput. Yet, interestingly, itturns out that one can, not only increase network throughput, butalso increase the average individual throughputs of nodes. Thisis best explained through the simple example networks shownin Fig. 1. Using these examples, we will also show that the Txrate needs to be reduced carefully, otherwise, it can lead to adeterioration of network performance. From now on, we willdenote a MAC protocol which uses joint rate adaptation andnetwork coding by RANC.

c. To code or not to code

Consider the network configuration in Fig. 1 which includesthree nodes and one relay. Node A transmits packets to nodeC and node B transmits packets to node A. Both transmissionsare relayed through node R. Let CXY denote the link capacitybetween node X and Y, and let CAR == CRA == CRC == 1bps.Suppose node A and B each transmit one packet to relay Rrespectively. Then, node R relays the two packets to node Cand A using either network coding or simple relaying. For eachcase, we will calculate the network throughput defined as the totalnumber of bits transported divided by the total amount of timeto transport the bits. For simplicity, we assume the instantaneousTx rate on a link is given by its link capacity. We also assume

1We consider fixed Tx power.

that if node B transmits at rate CBC to node C, then any node Xwith link capacity C BX > CBC can overhear the transmission.

In Case (a) shown in Fig. 1 where CBR == 0.5 and CBC == 1,node B transmits with rate CBR and node C can overhearthe transmission since node B is closer to node C than tonode R. In this case, node R can perform a bit-by-bit XORover the two packets from node A and B and broadcast. Intum, node C receives the coded packet and can decode it sinceit has the overheard the packet from node B. The network

throughput for this case is TJ:'b == Q+l+Q == ~, where the 2bin numerator is the number of bits 1tr~Jspbrted in the networkand ~ + O~5 in denominator is a time required to send twopackets from source nodes A and B to relay R. The additionalterm, ~ in the denominator is the time required for the relay tobroadcast the combined packet. In a similar way, the networkthroughput when only relaying is used can be calculated and. T(a) 2b 2 CI I h h·IS R == Q+.J....+Q+Q == S· ear y w en over eanng occursnaturally, inler-~~ssionl coding increases network throughput.

In Case (b) where C BR > CBC, overhearing no longer comesfor free. Unless node B decreases its instantaneous Tx rate, it cannot ensure node C overhears its transmissions. Once it reducesits instantaneous Tx rate (e.g, by reducing the modulation orderor decreasing channel coding rate) node C will overhear thetransmission and node R can perform network coding. Suppose,C BR == 1 and CBC == 0.8. Then, if B reduces its rate to R from1 down to 0.8, we have TJ:b == !3, otherwise we have Tj;) == 1.The benefit of rate reduction should be clear in this case. Hence,for both Case (a) and (b), network coding is beneficial.

Our last Case (c) is different. In this case, even though we canensure node C overhears by reducing node B's instantaneousTx rate, this will not increase the network throughput since B'stransmission would be excessively slow. In this case, relayinggives higher network throughput: Tt-b == g< T~c) == 1. Theseexamples show that the TX rate should be carefully determined.

In Fig. 2, we illustrate the potential locations where instanta­neous rate adaptation is beneficial for the simple network topologydiscussed above. To generate these figures we assumed, a staticfree space channel model with attenuation factor of 3.5 withoutfading and shadowing were considered. A transmit power of 1Wand noise power spectral density of -174dBm were used. Theminimum required SNR for decoding was set to as 3dB. TheTx rate of any link was set to the Shannon capacity of the link.Fig. 2a and Fig. 2b show what is the optimal relaying strategyfor all possible locations for node B. Here, nodes A, Rand Care fixed and we move node B to all possible locations in a two

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150 200 250 300

Jl(m...)~ ~ ~ ~ ~ ~ ~ ~ ~

x(melor)

o~_-----

o ~ ~ ~ ~ ~ ~ ~ Gx (moter)

(a) (b) (c) (d)

Fig. ~: Optimal r~laying strate~ for the four n~es n~twork in Fig. ~ as the position of node B is varied: (a) Optimal relaying method in a network with regular network coding, (b) Optimalr(~!a)ymfg methdod l.n a network Wlt~ network codmg wIth rate adaptatIOn, (c) Throughput gain (%) of rate adaptive network coding compared to without network coding (d) Throughput gain

;10 0 rate a aptlve network codmg compared to regular network coding. '

dimensional square of size 400 x 400m2 . The distance betweennodes A and R is equal to that of node Rand C - 80 meters.For any given location for node B, we first check whether nodeB can communicate with either node R or A. If node B cannot communicate with either of them, then, we declare it "notreachable." If node B can communicate with either of themthen, three possible relaying strategies were compared: direc~delivery, simple relaying, and network coding. The throughput iscomputed as previously. The best relaying method, i.e., giving thehighest throughput, was selected. One can visualize the expandedregion where joint rate adaptation and network coding are usedby comparing Fig. 2b to Fig. 2a.

Fig. 2c exhibits throughput gain of rate adaptive networkcoding compared to direct relaying method. Clearly the gaindepends on node B's location. We get the highest gain whennode B is at the midpoint of the line connecting nodes RandC, since at that location node B can send at its highest ratewithout rate reduction. Note that for network coding, node Bshould make sure that both nodes Rand C hear the transmission.Fig. 2d exhibits the gain of network coding with rate adaptationversus regular network coding without rate adaptation. One cansee that the region where the gain is positive is the same asthe expanded region in Fig. 2b. Depending on the location ofnode B, one can expect various gains upto 33%. Note that thisis an additional gain from rate adaptation for network coding.More generally one might expect this result to hold for networktopologies with more nodes around relay hubs where there aremore coding opportunities. In other words, it appears that rateadaptation should reduce the area of region where relaying anddirect delivery are better than network coding.

III. SYSTEM MODEL AND PROBLEM FORMULATION

A. Network Model

We consider a fixed wireless network having a relay z viawhich a set of source nodes S communicate with a set ofdestination nodes D. Each source node xES has a correspondingdistinct destination node y ED. A pair of source and destinationnode specify a session. Let s (y) denote the corresponding sourcenode of a node y ED. Each source node has only one desti­nation node and vice versa2. We assume an ideal MAC without

2This assumption can be justified by separating a physical source(destination)node with two different destination(source) nodes into logical twosource(destination) nodes.

contention, i.e., each source nodes take its turns to transmit itspacket. The number of bits per packet is fixed.

B. Transmission rate vector

We define R == (r i , 1 :::; i :::; IRI) to be an ordered set ofdiscrete rates supported by nodes, where IRI < 00 denotes thecardinality of the set R. So, r 1 and rl'Rl are the lowest and thehighest supported rates, respectively. The assumption that R is afinite set is intended to model today's systems, where the numberof Tx rates supported is limited due to modulation, slot length,coding rate, etc [11] . Note that even though we have a rate setR, the actual Tx rate at any link is determined by the transmitpower, distance, thermal noise, bandwidth, interference, etc. Themaximum Tx rate between node x and y is formally defined as

r~ = max {r E Rlr ~ i log(l + pllx17-+YJI-a)} ,

where p is the Tx power of a node, TJ is noise power spectraldensity, w is system bandwidth under consideration, Q is theattenuation factor, II x - y II is distance between two nodes x and y,and I is the amount of external interference power. Losses fromimperfect measurements of interference power and thus incorrectrate adaptation will be reflected as a packet error probability3.So, the effective Tx rate is given by r"; times the packet deliveryprobability between x and y. For xES, let

R x == {min(r~,r~)ly ED U {z}}

be the set of the highest rates from node x to relay z that allowat least one node y E D U {z} including z to hear node x'stransmission. This can be understood as the set of node x'spossible highest Tx rates to relay z which allow overhearing byothers. The objective is to determine a vector r == (rxz : XES)of Tx rates that each node will use to transmit to the relay. Wehave at most IlxEs IRx I such vectors since each node xES hasIRx I different potential Tx rates.

C. Overhearing Graph and Clique Partition

For each Tx rate vector r, a set of destination nodes thatoverhear the transmission can be determined, which then alsodetermines which packets the relay node can combine. Based on

3In the evaluation conducted Section V, we ignore interference I since weconsider a star topology network without contention.

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the overhearing status of each destination node, we construct agraph G(r) such that any two destination nodes which overheareach other's source node are connected by an undirected edge.We identify sets of packets that can be coded together with validpartitions V (r) of the graph. These are explained below.

The graph G(r) = G(D, Er ) is composed of a set of des­tination nodes D and a set of undirected edges Er . The link(i, j) E D 2 is in E r if and only if node i overhears node s(j) 'stransmission and node j overhears node s(i) 's transmission. Wesay node i overhears node s(j) if node i is in the transmissionrange of node s(j), or equivalently if r;(j)i ~ rs(j)z' So given aTx rate vector r uniquely determines an overhearing graph. Theset of links Er is formally defined as

Er = {(i,j) E D2 Ir;(i)j ~ rS(i)Zl r;(j)i ~ rs(j)z} .

We use this overhearing graph to find optimal sessions to com­bine. Note that, by construction, sessions corresponding to nodesin any valid clique in G(r ) can be combined because eachdestination node in the valid clique can overhear other destinationnodes' source nodes. So, finding a set of sessions to combinecorresponds to finding a clique and finding a family of setsof sessions to combine, thus, corresponds to finding a partitionV = (C1, C2 , ... ) of G(r) such that each set Ci of the partitionis a clique in G(r). We shall refer to such partition as a clique­partition. Note that there may be one or more valid clique­partitions in the graph. We let V (r) be the set of all valid clique­partitions of G(r).

D. Cost Function and Formulation

For a given r and V E V (r), we define the uplink cost as a totaltime required for all source nodes to transmit their packets to therelay, i.e., u(r) == EXES ~. The uplink cost is a function of ratevector r. Similarly, the do~nlink cost is a total time required forrelay to send the received packets to corresponding destination

nodes, i.e., d(V) == ECEv miny~c Tz

' where for. each C E Vwe take the minimum rate of rzy fbr y E C SInce we wantto ensure all destination nodes of the combined packets receivetheir associated packets. Note that downlink cost depends on theselected combination of sessions or clique partion V E V (r ). Ourobjective is to find an optimal rate vector r* and an associatedclique partition of G(r) which minimize the sum of the uplink andthe downlink cost, i.e., maximize a throughput. This optimizationproblem is formally stated as follows:

1 1min L - +L' . (1)rER rxz mlnyEC rzy

VEV(r) xES CEV

Let r* denote a solution of the above problem. This combinatorialoptimization problem can be decomposed as follows. For a givenr E R, we first need to find the minimum downlink cost in orderto evaluate the total cost for the r. Let V; be the optimal cliquepartition giving the minimum cost of the downlink under the ratevector r. Then, our original problem can be rewritten as follows:

V; E arg min d(V). (3)VEV(r)

Finding the optimal partition V; is a variant of well known cliquepartition problem (CPP). CPP partitions a graph G into disjointcliques with a minimum number of cliques. Determining theminimum clique partition is known to be a NP-hard problem.Indeed problem (3) is reduced to the classical CPP if cost foreach clique is equal to 1, see [12], [13]

Remark: Note that the decreasing the Tx rate for uplinktransmission makes it easier for each node to overhear other trans­missions, which makes the overhearing graph more connected.This in principle results in an increase of the possible codedtransmissions, which may ends up decreaseing the downlinkdelays. So, we see the tradeoff between the uplink and thedownlink delay. Our objective is to find optimal point where sumof those two delays are minimized.

E. Pairwise Coding

For simplicity, suppose we restrict the space of clique partitionsV (r) to partitions including cliques of size less than or equal totwo, i.e., we determine only an approximate clique partition, V~*.In this case, (3) can be rewritten as the following binary integerprogram:

1min L - + L alCl

al,lEEr rzy l EyED E r

s.t. L al:::; 1, Vy E DlE{ (y,i)EEr liED}

alE{O,l}, VlEEr

111Cl = - -- - -- Vl E Er , (4)

min {rZYI (l), r zY2(l)} r ZYI (l) r zY2(l)

where Yl (l) and Y2 (l) are the two nodes at two end points ofundirected link l. Also al is binary variable for link lEEr,

if packets destined to Yl (l) and Y2 (l) then al = 1, otherwiseal = O. Our restriction makes the sub-problem (3) easier. In factthe maximization version of (4) corresponds to the well knownmaximum weighted matching problem (MWMP). MWMP findsa maching M c E of graph G = (V, E) such that the sumof weight of 1 E M is maximized. The MWMP is the first"true" binary integer programming problem with polynomial timealgorithm4, see [14]. The linear programming relaxation of (4)with additional constraints is given as follows:

1min L - + L alCl

al,lEEr D rzy l EyE E r

s.t. L al ::; 1, Vy E D,lE{ (y,i)EErliED}

L al < ll!flJ V odd sets H ~ DlEE(H) - 2

where V; is given as

min u(r) + d(V;),rER

(2)4It is "true" binary integer problem in the sense that it can not be solved mearly

by linear programming relaxation.

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where Cz is defined in (4), E(H) is the set of edges with bothends in H. Note the role of additional constraints called odd setconstraints, odd set is a set with k nodes, for k == 3, 5, 7, .... Theconstraints restrict the number of pairs in any odd set H ~ Dbe less than or equal to lIHI/2J, i.e., it prevents odd cycles.The LP-relaxation without this restiction may not result in integersolution. (a)

+---+-----+--+-__+_L

(b)

IV. HEURISTIC ALGORITHM FOR RATE SELECTION

A. Assumption

We assume that any source node in this network knows theaverage link rate between itself and other reachable destinationnodes including the relay. These might be estimated by observingRTS and CTS signals or pilot signals. This information is sharedby each source node with the relay node, which in tum runsfollowing algorithms to perform joint rate adaptation and networkcoding.

B. Suboptimal Rate Selection Algorithm

In this section, we provide a heuristic algorithm to findsub-optimal rate vector f*, for problem (2). Note that thereare fIXES IRx I possible Tx rates vectors. Rather than doingan exhaustive search, we shall perform our search as follows.First, note the relation between the Tx rate vector and theoverhearing graph in a given network. The maximum Tx ratevector r m == (r~ IXES) results in an overhearing graphG(rm

) == G(D,Erm), with a minimum number of edges. Notethat this is a subgraph of all possible overhearing graphs thatcan be generated for all feasible Tx rate vectors. At the otherextreme if r l == (rxz == rllx E S), then the corresponding graphG(r l

) == G(D, Erl) is the supergraph of all possible overhearinggraphs in this network. So, these two graphs are at extreme ends.For r =I- r m , r l

, the set of edges E r satisfies Erm ~ E r ~ Erl.Our basic strategy is to consider sequence of vectors rl, r2, ... ,such that Erm ~ E r1 ~ E r2 ~ ... ~ Erl. We shall do this bygradually decreasing the Tx rates by limiting individual Tx ratewith Tx rate bar L. Initially, L is set to the highest rate r lnl ,and gradually decreased. For a given L, the transmist rate fromsource to relay may not be feasible, so, the Tx rate of a sourcenode xES is set to min{r~, L}. With this choice of rates wecan quickly estimate the value cost function.

1) Clique Partition and Cost Evaluation: Every time welower Tx rate bar L, a new graph which possibly includesadditional edges is produced. Then, a pairwise clique partitionof the graph is found by solving (5). Based on this cliquepartition, we can evaluate the corresponding downlink cost. Toevaluate uplink, we need to only optimize over Tx rates fromthe sources S to relay that result in the same overhearing graphand thus the same downlink cost. If a clique is of size two, e.g.,{Yb Y2}, then, r s(yI)z is be choosen as high as possible, i.e.,min{r:(yI)z' r S(yI)Y2}' Note that if r s(Ydz is increased more thanthat, then, one can not ensure Y2 will overhear its transmission,and the clique is broken. The same applies to Y2. If a cliquesize is one, e.g., {y}, rs(y)z, is reverted to its original max rater:(y)z' which may be higher than L. This procedure determinesthe minimum uplink cost.

(c)

Fig. 3: Tx rate bar lowering and corresponding transfonnation of overhearing graph in thenetwork with four sessions: (a) Network with four sessions, (b) Max Tx rates of four sourcenodes and Tx rate bar L. The actual Tx rate of node X i is detennined as min(r ~ z , L).(c) As Tx rate bar L decreases from top to bottom overhearing graph (4 nodes blac~ edgesbetween them) changes from (1) to (6) in order. Corresponding optimal pairwise cliques (indotted red) also change.

2) Choosing Sub-optimal Rate Vector f*: Every time theTx rate bar decreases a step, a total cost is evaluated. Thiscontinues until the bar hits the bottom of the supported rate setR. At this point the rate vector value resulting in the minimumcost is selected as the approximation of the optimal Tx ratevector f*. This algorithm is formally described in Alg.l, whereFindCliquePartition is a function which takes the rate vector r asan input and produces optimal pairwise clique partition Vf* as itsoutput, i.e., it solves problem (5), and K is parameter dependingon relay's access opportunity.

3) Example: The algorithm is explained with the examplenetwork shown in Fig. 3a. We have four source sessions andcorresponding maximum Tx rates from source nodes to relay z:r~z < r~z < r~z < r~z' The Tx rates from the relay to thedestination nodes are given as r ZY4 < r ZY2 < r ZY3 < r ZYI' Theset of supported Tx rates R is defined as {I, 2, .. ·8}.

To check whether there exists any better choice for the Tx ratevector, we first set the Tx bar L to 8 and compute overhearinggraph. This graph is shown as (1) in Fig. 3c. Without rateadaptation, node YI and Y4 are overhearing with each other'ssource nodes so they can be combined. While, node Y2 and Y3can not be grouped. If L is lowered below 6, so that the Tx rate ofnode Xl and X4 are limited by L, then, we can make sure that Y3overhears X4 's transmission. This introduces a new link betweenY3 and Y4. When the clique partition is found over this graph, Y4is paired with Y3 rather than YI since r ZY3 < r ZYI' If we furtherdecrease L to 4, so the Tx rate from X2 is limited by L, then,Y4 overhears X2 's transmission, which adds a new link betweenY4 and Y2 as shown in (3) of Fig. 3c. Again we can performclique partitioning over the graph and evaluate minimum cost. Ina similar way, if we continue to lower L to 1, then, we obtain thecomplete graph (6) in Fig. 3c and corresponding clique partition.Among all Tx rates evaluated, the one giving minimum cost isselected. Note that the L is the control parameter for the tradeoffbetween the two costs.

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Algorithm 1 Sup-optimal Rate Selection Algorithm

1: i {= Inl2: while i > 0 do3: L {= r i

4: rxz {= min{r~,L},VxE S5: Vf* {= FindCliquePartition (r)6: rs(j)z {= maxYECU{z}\{j} rr;(j)y' Vj E C, VC E Vf*7: ri

{= (rxz Ix E S)8: £(ri ) {= ~ K +~" 1

L.JxES Txz L.JWEV minYEW Tzy

9: i {= i-I10: end while11: f* {= arg min l:O::;i:O::;lRI £(ri

)

v. PERFORMANCE EVALUATION

A. Simulation Environment

In this section, we evaluate the performance of a network usinga joint rate adaption and network coding scheme. In particular,we consider a star topology network where a relay node receivespackets from source nodes and transmits them to destinationnodes. Source and destination nodes are randomly placed withinthe Tx range of the relay such that the source and destinationnode can not directly communicate with each other. Each nodecan support 12 Tx modes, with different modulation and codingrates, transmitter chooses the highest Tx rate supportable basedon the received average SNR at the receiver. We say a Txmode is "supportable" if a desired target PER is achieved underAWGN channel for given average SNR. Static AWGN channelwith a path loss attenuation factor of 3.5 was considered. Weconsider two extreme scenarios, in which relays have differentMAC opportunities. In the first scenario, we allow the node actingas relay have higher access opportunity than neighboring nodesacting as data sources. This policy allows us to evaluate thepure gain of network coding, which is not affected by MAC.While, in the second scenario, we assume all nodes includingthe relay have equal access to the medium. This case showshow network performance is affected by joint rate adaptation andnetwork coding and how these interact with the MAC. Note that,in both cases, we give access opportunity to each source nodeunder max-min fair policy in long term average Tx rate. The onlydifference is the relay's access opportunity. As a performancemetric, network throughput is calculated as the number of bitstransported divided by the amount of time spent on uplink anddownlink packet transmissions.

B. Case 1: Relay with more access opportunity than other nodes

In this case, each node takes its tum to transmit a packet(uplink) to the relay and then the relay consumes as manyTx opportunities as it needs to serve the received packets ondownlink to destination. The uplink and downlink transmissionsform one cycle, which repeats. It keeps max-min fairness onlong term average rate across source nodes (K == 1). Note thepredetermined Tx orders removes contention, and accordinglythere is no throughput loss from the contention for the medium.This allows to study the network which achieves its maximum

throughput. In this sense, the gains of joint rate adaptation andnetwork coding over simple network coding or no coding at all,can be viewed as pure gains.

Fig. 4 and 5 show an average throughput and an averagegain of the rate adaptive network coding (RANC2) and regularnetwork coding (RNC2) both using only pairwise coding policyover random star network topologies. As the number of sessions,IN I, increases, the coding opportunities at the relay increaseresulting in a throughput increase. The throughput gain ofRANC2compared to no network coding case ranges from 9% to 19% andthe gain compared to RNC2 is around 4% as INI ranges from 2to 8. Note the gains of baseline or pairwise network coding, looksmarginal. This is due to the throughput averaging over randomnode placement. So, the average gain reduces from the maximumgain of33% to 6-15% depending on INI. In other words, networkcoding is not helpful for substantial node placements. In fact,we observed 40% of node placements has no coding opportunityunder RNC2 when INI == 2. For RANC2, these cases reduces,but the gain looks still marginal.

C. Case 2: Relay with equal access opportunity as other nodes

In our second scenario, we assumed that all nodes includingthe relay have equal access to the medium. That is, the relay,as other nodes, has only one Tx opportunity per cycle. So, therelay behaves like a bottleneck node, in which unsent packets aredropped immediately. So, the relay is supposed to choose onecoded or non-coded packet each tum that it gets so as to keep

*fairness in the long run Tx rates across sessions (K == IV/I).This scenario gives an idea on the performance of joint rateadaptation and network coding in a heavily loaded networks. Notethat all nodes in such a network are assumed to be backloggedand have equal access opportunities.

Fig. 6 and 7 shows the throughput and the thgoughput gainof RNC2, RANC2, RNC and RANC under this scenario, whereRNC and RANC are regular and rate adaptive network codingwithout a limit on the number of packets combined. We firstobserve that the throughput decreases as INI increases. It's thenatural result of our assumption. As INI increases, the number ofdropped packets also increases and the throughput decreases. Thegain of the regular pairwise network coding RNC2 to no codingat all ranges from 20-42%. Once a rate adaptation is applied, thegain increases from 40-62%. However, the relative gain (RANC2to RNC2) gradually decreases. This is because as the the numberof nodes increases, packets are easily paired with another packeteven without rate adaptation. Even though pairwise coding looksquite effective as compared to previous scenario, it still does notfully resolve the congestion at the relay.

This motivates us to introduce a more general clique partition­ing scheme. For this purpose, a new simple sub-optimal cliquepartitioning method is introduced at Appendix. Note that thecoding with high degree resolves the congestion, and increasesthroughput. The gain (RANC to noNC) ranges 40% to 120%.And, the relative gain to RNC2 also increases as the number ofnodes increases. Note that this implies that increasing the allow­able coding degree effectively decreases the number of packetsdropped, which is directly translating to throughput increases.

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• n " .

Fig. 6: The average throughput when the relay have equal access priority

9

•

"noNe+RNC2.. RANC2".RNC~RANC

7

..RNC2 to noNe

... RANC2 to noNe"'''RANC2toRNC

••

456Number of nodes

456Number of nodes

456Number of nodes

2O.....---------------~

1

O'-----------------....J1

36

35

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~32.c.C)

g 31.c.... 30 ........

29

281

30r-----------------.....25

5

20r-----------------....18

16

14

_12oe.~10'iiiC) 8

6

en 20a.eg, 15.c.OJ::lo.c 10....

Fig. 4: The average throughput when the relay has higher access priority

Fig. 5: The gain average throughput when relay has higher access priority

ApPENDIX

A heuristic clique partitioning algorithm based on [15].

REFERENCES

VI. CONCLUSION

In this paper, we studied the impact of rate adaptation forinter-session network coding for star topology wireless network.We showed that rate adaptation can be effective at increasingnetwork coding gain and in resolving severe congestion at relay,in particular, in networks where standard MAC protocols are usedin which all nodes have equal access opportunities.

[1] R. Ahlswede, N. Cai, S. Y R. Li, and R. W. Yeung, "Network infonnationflow," IEEE Trans. Infomation Theory, vol. 46, no. 4, pp. 1204-1216, 2000.

[2] S. Y. R. Li, R. W. Yeung, and N. Cai, "Linear network coding," IEEETrans. Infomation Theory, vol. 49, no. 2, pp. 371-381, 2003.

[3] R. Koetter and M. Medard, "An algebraic approach to network coding,"IEEE/ACM Trans. Networking., vol. 11, no. 5, pp. 782-795,2003.

[4] D. S. Lun, M. Medard, and R. Koetter, "Efficient operation of wirelesspacket networks using network coding," IWCT, 2005.

[5] D. S. Lun, M. Medard, and M. Effros, "On coding for reliable communica­tion over packet networks," Proc. ofAllerton Conf. on Comm. Control andComp., Sep 2004.

[6] P. Chou, Y. Wu, and K. Jain, "Practical network coding," Proc. ofAllertonConf. on Comm. Control and Comp., 2003.

[7] Sachin Katti, Hariharan Rahul, Wenjun Hu, Dina Katabi, Muriel Medard,and Jon Crowcroft, "Xor in the air: practical wireless network coding,"ACM SIGCOMM, 2006.

[8] 1. Liu, D. Goeckel, and D. Towsley, "Bounds on the gain of network codingand broadcasting in wireless networks," Proc. IEEE INFOCOM, 2007.

[9] 1. Le, 1. C.S. Lui, and D.M. Chiu, "How many packets can we encode? ­an analysis of practical wireless network coding," Proc. IEEE INFOCOM,2008.

[10] A. K. Haddad and R. Riedi, "Bounds on the benefit of network coding:throughput and energy saving in wireless networks," Proc. IEEE INFOCOM,2008.

[11] "Ieee 802.11 a standard," .[12] Dion Gijswijt, Vincent Jost, and Maurice QueYranne, "Clique partitioning

of interval graphs with submodular costs on the cliques," Tech. Rep. TR­2006-14, Egervary Research Group, Budapest, 2006, www.cs.elte.hu/egres.

[13] Saul G. de Amorim, Jean-Pierre Barthelemy, and Celso C. Ribeiro, "Clus-tering and clique partitioning: Simulated annealing and tabu search ap­proaches," Journal of Classtfication, 1992.

[14] George L. Nemhauser and Laurence A. Wolsey, Integer and CombinatorialOptimzation, WILEY, 1988.

[15] long Tae Kim and Dong Ryeol Shin, ''New efficient clique partition­ing:algorithms for register-transfer SYnthesis of data paths," Journal of theKorean Physical Society, vol. 40, no. 4, pp. 754-758, April 2002.

... RANC to noNC ...'RANC to RNC2

..RNC2 to noNe + RANC2 to noNe .... RANC2 toRNC2

140

120

100

t 80

c'iii 60C)

40

20

01 456

Number of nodes

Fig. 7: The gain of average throughput when the relay has equal access priority

Algorithm 2 Sup-optimal Clique Partition Algorithm1: N~D

2: while N i= ¢ do3: Select pEN with minimum degree. Tie breaking: select

p with minimum tx rate from relay to p.4: Select a node q, a neighbor of p, with the maximum com­

mon edges with p. Tie breaking: select q with minimumTx rate from relay to q

5: Delete edges from p and q that do not connect to theircommon neighbor.

6: Merge p and q and rename it as p with transmission ratemin{rzp , rzq }.

7: Ifp has any remaining edge goto step 4 otherwise p becomea new clique. Remove p from N.

8: end while

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